If $R = \{(6, 6), (9, 9), (6, 12), (12, 12), (12, 6)\}$ is a relation on set $A = \{3, 6, 9, 12\}$,then relation $R$ is

Let the relation $R_{1}$ be defined on $R$ as $a R_{1} b$ if $1+ab > 0$. Then

If $R$ and $R^1$ are equivalence relations on a set $A$, then which of the following is also an equivalence relation?

Let $L$ be the set of all lines in a plane and $R$ be the relation in $L$ defined as $R = \{(L_{1}, L_{2}) : L_{1} \text{ is perpendicular to } L_{2}\}$. Show that $R$ is symmetric but neither reflexive nor transitive.

Let $A = \{1, 2, 3\}$. Show that the number of relations on $A$ containing $(1, 2)$ and $(2, 3)$ which are reflexive and transitive but not symmetric is $3$. (Note: The original prompt claimed $4$,but the correct count for this specific set is $3$).

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(x, y) \mid x, y \in R \text{ and } xy \text{ is an irrational number}\}$. Then,the relation $r$ is: